https://neckpinch.com/authors/xinrui-zhao/New arXiv papers by Xinrui Zhao, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2607.23879arxiv:2607.23879Sun, 26 Jul 2026 00:00:00 GMTJingwen Chen, Tang-Kai Lee, Ao Sun, Xinrui Zhao — Given an embedded shrinker Σ in ℝⁿ⁺¹ that is either closed, asymptotically conical, or a Cartesian product of such a shrinker with ℝ^k, we construct a closed embedded mean curvature flow whose tangent flow at the first singularity is modeled on Σ. We also prescribe the first-order asymptotics of the tangent flow. This result is a consequence of a more general theorem that allows us to construct mean curvature flows with an additional force whose tangent flow and first-order asymptotics at the first singularity are prescribed.Mean curvature flowRicci solitonsSingularitiesBoundaries & non-compact https://arxiv.org/abs/2510.23499arxiv:2510.23499Mon, 27 Oct 2025 00:00:00 GMTYiqi Huang, Xinrui Zhao — We prove that if a rescaled mean curvature flow is a global graph over the round cylinder with small gradient and converges super-exponentially fast, then it must coincide with the cylinder itself. We also show that the result is sharp with counter-examples of local graphs at arbitrarily super-exponential convergence rate with the domain expanding arbitrarily fast. The first part provides the first unique continuation result in the cylindrical setting, the generic singularity model in mean curvature flow. In sharp contrast, in the second part we construct infinite-dimensional families of Tikhonov-type examples for nonlinear equations, including the rescaled mean curvature flow, showing that unique continuation fails for local graphical solutions. These examples demonstrate the essential role of global graphical assumptions in rigidity and highlight new phenomena absent in the compact case. We also construct non-product mean curvature flows that develop singular sets as prescribed lower dimensional Euclidean space at arbitrary super-exponential rates. Our construction works in great generality for a large class of non-linear equations.Mean curvature flowSingularitiesStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2405.15577arxiv:2405.15577Fri, 24 May 2024 00:00:00 GMTTang-Kai Lee, Xinrui Zhao — In this paper, we prove that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. The main technique is the Ważewski box argument, used by Stolarski in the proof of the corresponding theorem in the Ricci flow case. As a corollary, our construction, combined with the works of Angenent–Ilmanen–Velázquez and Chodosh–Daniels-Holgate–Schulze, implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. These provide examples related to a question asked by Evans–Spruck.Ricci flowMean curvature flowRicci solitonsSelf-shrinkers & translatorsSingularitiesWeak & singular flowsBoundaries & non-compact https://arxiv.org/abs/2301.10883arxiv:2301.10883Thu, 26 Jan 2023 00:00:00 GMTTang-Kai Lee, Xinrui Zhao — In this paper, we prove the uniqueness of asymptotically conical tangent flows in all codimensions. This is based on an early work of Chodosh-Schulze, who proved the uniqueness in the hypersurface case.Mean curvature flowSingularitiesStability & uniquenessBoundaries & non-compact